Overview of the group isomorphism theorems

From Maths
Revision as of 00:21, 22 July 2016 by Alec (Talk | contribs) (Alec moved page Group isomorphism theorems to Overview of the group isomorphism theorems: It's an overview.)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Saving work, will come back to later

Statements

[ilmath]\xymatrix{G \ar[r]^\varphi \ar[d]_\pi & H \\ G/N \ar@{.>}[ur]_{\bar{\varphi} } & }[/ilmath]

Group factorisation theorem

[ilmath]\xymatrix{G \ar[r]^\varphi \ar[d]^\pi & H \\ G/\text{Ker}(\varphi) \ar@{.>}[r]_-\theta & \text{Im}(\varphi) \ar@{_{(}->}[u]^i }[/ilmath]
For [ilmath]\theta[/ilmath] an isomorphism
First isomorphism theorem

First group isomorphism theorem (AKA: Group homomorphism theorem)

[ilmath]\xymatrix{G \ar[r]^\varphi \ar[d]_\pi & H \\ G/N & }[/ilmath]

Second group isomorphism theorem

[ilmath]\xymatrix{G \ar[r]^\varphi \ar[d]_\pi & H \\ G/N & }[/ilmath]

Third group isomorphism theorem

References